Computation of displacements from strain fields: Derivation, validation, and application

High-resolution experimental methods now yield detailed elastic strain fields around cracks and other stress concentrators; however, these measurements alone cannot provide the displacement gradients required for a quantitative fracture assessment. Existing approaches to recover displacement fields from such data are either restricted to two dimensions, rely on strong assumptions about material behaviour, or lack robustness when confronted with noise typical of diffraction-based strain mapping. To address this limitation, we have developed a finite-element framework that reconstructs continuous displacement fields directly from measured deformation gradients without prescribing external loads or boundary conditions. The approach formulates strain integration as an over-determined least-squares problem defined on linear or quadratic elements in two and three dimensions. Validation using analytical 2D mode-I and 3D mixed-mode crack-tip fields demonstrates that the recovered displacements and derived stress intensity factors deviate from theoretical values by <4 %. The method is further applied to HR-EBSD measurements around Vickers-indentation cracks in monocrystalline silicon, where the reconstructed fields reproduce AFM-measured surface topography and reveal distinct residual mixed-mode loading at neighbouring cracks. These results show that full-field diffraction-based strain measurements can be converted into mechanically meaningful displacement and fracture parameters with high fidelity. The method provides a practical route for quantifying crack driving forces in complex microstructures and can be extended to three-dimensional diffraction techniques and in situ studies of evolving damage.